I have a way of constructing (something like) a 2-category from a category with products $\mathcal{C}$.

**My question**: Is this a correct construction, and if so, does it have a name?

Define $\mathcal{C'}$ to consist of one object. The 1-morphims are objects of $\mathcal{C}$, the identity is the terminal object of $\mathcal{C}$, and $A \circ B$ is $A \times B$. The 2-morphisms $A \to B$ are morphisms $A \to B$ from $\mathcal{C}$. Vertical composition is composition in $\mathcal{C}$ and the horizontal composition of $f$ and $g$ is given by $f \times g$.

Notes:

I said "something like" a 2-category because it seems like composition will be defined up to isomorphism, rather than equality.

Perhaps this would work with any monoidal category, rather than just a category with products.

I want this construction so I can "reverse" the usual relationship between monoid object and monad, and say that a monoid in $\mathcal{C}$ is given by a monad in $\mathcal{C'}$.